Statistics of Lyapunov exponent in 1-D random periodic-on-average systems

TL;DR

This paper investigates how classical waves localize in one-dimensional random periodic systems, revealing two distinct localization modes and the conditions under which universal scaling behavior occurs.

Contribution

It demonstrates the existence of two localization regimes in 1D random periodic systems and characterizes their scaling properties using Monte Carlo simulations.

Findings

States in pass bands show universal single parameter scaling.

States in band gaps require two parameters for scaling.

Transition between behaviors occurs in a narrow frequency region.

Abstract

By means of Monte Carlo simulations we show that there are two qualitatively different modes of localization of classical waves in 1-{\em D} random periodic-on-average systems. States from pass bands and band edges of the underlying band structure demonstrate single parameter scaling with universal behavior. States from the interior of the band gaps do not have universal behavior and require two parameters to describe their scaling properties. The transition between these two types of behavior occurs in an extremely narrow region of frequencies. When the degree of disorder exceeds a certain critical value the single parameter scaling is restored for an entire band-gap.